Superconvergence in free probability limit theorems for arbitrary triangular arrays

TL;DR

This paper extends superconvergence results in free probability limit theorems from identically distributed arrays to arbitrary triangular arrays, showing density convergence at points where the limit density is nonzero.

Contribution

It demonstrates that superconvergence in free probability holds for non-identically distributed arrays, broadening the scope of previous results.

Findings

Superconvergence applies to arbitrary triangular arrays.

Density convergence occurs at points with nonzero limit density.

Extends known results beyond identically distributed cases.

Abstract

It is known that limit theorems for triangular arrays with identically distributed rows yields convergence of densities rather than just convergence in distribution. We show that this superconvergence result holds -- at least at points at which the limit density is nonzero -- even if the rows of the array are not identically distributed.